Some characteristics of supersonic flow of an electrically conductive gas in an MHD-channel

We study supersonic flows of an electrically conductive gas in crossed electric and magnetic fields [1] in the presence of shock waves. It is shown that three steady flow regimes can exist, and that these are defined by the electrical conductivity of the gas as a function of temperature and density.

1. The normal regime is characterized by a tendency for the shock to move toward the channel entrance on increase of the static pressure at the channel exit. The steady regime of this type exists and is stable.
2. The anomalous regime (formally constructed) is characterized by a tendency for the shock to move toward the exit on increase of the static pressure at the channel exit. This regime is unstable and the flow in the MHD-channel may be either entirely supersonic or entirely subsonic.
3. The limiting (boundary) regime is intermediate between the normal and anomalous regimes and is characterized by the fact that the stationary position of the shock wave and its amplitude are not uniquely defined. Steady flow in this case is not unique.

This study involves formal construction both of the solution to the steady-state problem and the corresponding nonsteady-state problem [4]. The establishment of a steady regime in the solution of the unsteady problem, is at the same time, a verification of its stability.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

超声速气流磁流体加速初步实验研究

利用激波风洞, 采用氦气驱动氩气, 在平衡接触面运行方式下得到高温气体,通过在低压段注入电离种子K₂CO₃粉末, 实现高温条件下导电流体的产生, 设计了超声速喷管及磁流体加速实验段, 采用大电容提供电能, 开展了超声速气流磁流体加速初步实验研究. 典型实验条件下, 当喷管入口总压为0.7049MPa、理论平衡温度为8372.8K, 喷管出口马赫数为1.5, 电容充电电压为400V, 磁感应强度为0.5T时, 对电压电流特性、电导率、负载系数、电效率、加速效果等进行了测量或计算, 主要结论有: 磁场作用下的超声速气流的电导率的值大约在150S/m; 磁流体加速通道负载系数约为4, 电效率约为28%, 平均输入功率约198kW; 采用电参数测试方法对磁流体加速效果进行评估, 速度增加约15.7%;超声速气流的电导率对加速通道的电效率及加速效果等有很重要的影响.      

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Reflection of magnetoacoustic waves

The problem of the reflection of magnetoacoustic waves at the boundary dividing an elastic medium from a fluid medium with infinite conductivity in the presence of an arbitrary constant magnetic field was treated in [1]. In writing down the boundary conditions the continuity of the tangential component of the magnetic field was used. This condition is valid when the conductivity of the medium is finite but not when the conductivity is infinite. In this connection a problem similar to that in [1] is solved, without employing this particular boundary condition. The amplitude conversion coefficients found for the limiting cases of weak and strong magnetic fields coincide with the respective coefficients given in [2,3] for media with a finite conductivity, if we allow the conductivity in the latter expressions to become infinite.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, January–February, pp. 56–61, 1970.     

Nonstationary flow of a conducting gas in crossed electric and magnetic fields

The plasma is inviscid, cool, and not thermally conducting; it flows in a channel of constant cross section. The solution is derived by the small parameter method, for which purpose the magnetic interaction N is used. There have been previous studies of the transient-state flow of an inviscid and thermally nonconducting plasma in crossed electric and magnetic fields [1–3]. A plasma of infinite conductivity has been considered [1], as well as flow involving entropy change in an MHD system with strong electromagnetic fields [2, 3].     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Effect of an inhomogeneous magnetic field on the structure of an ionized gas flow

Results of an experimental study of the flow of an ionized gas produced by a shock wave through an inhomogeneous magnetic field are presented. Braking of the gas flow produced by the end currents is determined at two fixed sections of the magnetogasdynamic channel from the value of the isolated shock wave formed in the vicinity of the hemispherical model over which the flow passes. Maximum recorded reduction in Mach number was 30%, and with a magnetogasdynamic interaction parameter greater than 1.5, a transition of supersonic flow to infrasonic at the exit of the magnetic zone was observed. Experimental results were compared with a solution of a model problem which assumed one-dimensional flow in the flow core. The gas used was argon, with a maximum magnetic field induction of 1.5 T.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–178, September–October, 1976.     

Features of the electric current and field distributions in flows with narrow layers of highly conductive gas

A solution obtained by Fourier's method provides the basis for analyzing the influence of a narrow gas layer, of higher conductivity than the rest of the flow, on the Joule dissipation and current distribution in the terminal zone of a plane magnetohydrodynamic channel with nonconducting walls. The MHD interaction parameter, Reynolds magnetic number, and Hall parameter are assumed small. It is shown that a narrow, highly conductive layer can on occasions be replaced by a surface of discontinuity, on which well-defined relations between the electric quantities are satisfied. The presence of such a layer leads to an increase in the Joule dissipation and a reduction in the lengths of the current lines. A hopeful arrangement for a magnetohydrodynamic energy converter is one in which an inhomogeneous flow is used, consisting of a continuous series of alternating very hot and less hot zones [1,2]. For this arrangement, it is worth examining the influence of the stratified conductivity distribution of the working body on the Joule dissipation and the electric currents in the channel. Numerous papers have discussed the case of inhomogeneous conductivity in the context of MHD system electrical characteristics. A general solution was obtained in [3] for the stationary problem on the electric field in a plane MHD channel with nonconducting walls when the magnetic field and conductivity are arbitrary functions of the longitudinal coordinate. In [4], where the braking of undeformed conducting clusters was investigated, the Joule dissipation, linked with the appearance of closed eddy currents in the cluster as it enters and leaves the magnetic field, was evaluated. The relationships between the electrical quantities, on moving through a narrow layer of low-conductivity liquid, were considered in [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 1, pp. 39–43, January–February, 1970.In conclusion, the author thanks A. B. Vatazhin for valuable advice and discussion.     

Two-dimensional flow of a perfect conducting gas in crossed electric and magnetic fields

Reference [1, 2] give a solution of the problem of the two-dimen-, sional flow of an inviscid thermally-nonconducting gas with constant conductivity in a channel of constant cross section for particular forms of the given applied magnetic field. The present paper obtains a solution of the problem of the two-dimensional flow of a gas with variable conductivity in crossed electric and arbitrary magnetic fields by means of the small parameter method. The magnetic Reynolds number R_m and the magnetohydrodynamic interaction parameter S are chosen as parameters. The international system of units is employed.Notation V flow velocity - j electric current density - p pressure in the flow - E electric field strength - gas density - electrical conductivity of the gas - T gas temperature - ratio of specific heats at constant pressure and volume - L channel half-height - ] permeability (magnetic) - B magnetic induction vector - B₀ applied magnetic field     

Stationary flow of a conducting liquid in a rectangular channel with two nonconducting walls and two walls having an arbitrary conductivity parallel to an external magnetic field

The flow of a conducting liquid in a channel of rectangular cross section with two walls (parallel to the external magnetic field) having an arbitrary conductivity, the other two being insulators, is considered. The solution of the problem is presented in the form of infinite series. The relationships obtained are used for numerical calculations of the velocity distribution and the distribution of the induced magnetic field over the cross section for several modes of flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostt i Gaza, No. 5, pp. 46–52, September–October, 1970.     

Convective instability of a current-carrying conductive liquid in a long channel with an external magnetic field

The stability of equilibrium of an electrically conductive liquid in a long channel in the presence of electric and magnetic fields is considered, under the condition that there exist gradients in temperature and liquid conductivity along the channel.     

Plasma flow accompanied by blowing and pumping of plasma across electrodes

The article discusses plane stationary slowly varying flows of a nonviscous plasma with good conductivity in a channel in a transverse magnetic field; the flows are accompanied by blowing in and pumping plasma across solid metallic electrodes. The Hall effect is taken into consideration. It is shown that the potential jump near the anode, which appears in an accelerated plasma flow in an ordinary channel with solid electrodes, can be eliminated in flows accompanied by blowing in (pumping) of plasma. It is also shown that flows are possible in which the velocity, density, and the transverse electric field increase in the direction of the accelerator cathode.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 26–34, November–December, 1970.     

Influence of magnetic reynolds number on the current distribution in a magnetohydrodynamic channel with allowance for the Hall effect

Stationary plane flow of a conducting gas across a magnetic field in a magnetohydrodynamic channel of constant cross section made up of electrodes of finite length and insulators is considered in the linear approximation. It is assumed that the electromagnetic forces are small. It is shown that the current density increases near the exit from the interelectrode gap with increasing magnetic Reynolds number. The mutual influence of the Hall parameter and of the magnetic Reynolds number on the distribution of the currents in the channel is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 148–152, May–June, 1971.     

On the possibility of blowing a gas jet into a supersonic flow without formation of a three-dimensional boundary-layer separation zone

We consider the flow formed by the interaction of a supersonic flow and a transverse sonic or supersonic jet blown at right angles to the direction of the main flow through a nozzle whose exit section is in a flat wall. When a gas jet is blown through a circular opening [1] the pressure rises in front of the jet because of the stagnation of the oncoming flow. This leads to separation of the boundary layer formed on the wall in front of the blowing nozzle. The resulting three-dimensional separation zone leads to a sharp increase in the pressure and the heat fluxes to the wall in front of the blowing nozzle, which is undesirable in many modern applications. The aim of the present investigation was to find a shape of the exit section of the blowing nozzle for which there is no three-dimensional separation zone of the boundary layer in front of the blowing nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–165, May–June, 1979.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Stability of subsonic gasdynamic flows

A system of differential equations describing small perturbations of the steady flow of a non-viscous ideal gas in a channel of variable cross section is analyzed in this paper. The equations of nonsteady flow and the boundary conditions are linearized, and the solution of the linearized equations is sought in the form v(x)expt t, where v(x) is an eigenfunction while is the natural frequency for the boundary problem being studied. With such an approach the problem is reduced to finding the solutions to ordinary differential equations with variable coefficients which depend on the parameter . Analytical solutions of this system are obtained for small values of and for values of ¦¦1. The results can be used to calculate the growth of high-frequency and low-frequency perturbations imposed on subsonic, supersonic, and mixed (i.e., with transitions through the velocity of sound) gasdynamic flows, to analyze the stability of subsonic sections, and to verify and supplement various numerical methods for calculating unsteady flows and numerical methods for studying stability in gasdynamics. The application of the solutions found for small and large is demonstrated on a study of flow stability behind a shock wave (a direct compression shock in the present formulation). Analytical expressions are obtained for the determination of from which it follows that the flow stability behind a shock essentially depends on the shape of the channel at the place where the shock is located in the steady flow, which was noted earlier in [1], and on the conditions of the reflection of small perturbations in the exit cross section of the channel, which was first pointed out in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–97, January–February, 1978.In conclusion, the author thanks A. G. Kulikovskii and A. N. Sekundov for helpful discussions of the work.     

Methods of calculation and numerical analysis of conducting-gas flow in high-current electric arcs

The mechanism of conducting-gas acceleration in an electric arc by intrinsic magnetic field was first investigated in [1]. Further theoretical study of this question was associated with the numerical calculation of arcs [2–7]. A more general approach to the solution of the problem was realized in [4], where the finite-difference method was used. Integral calculational models were developed in [5–7]. The present work proposes a modified version of the difference method [4] and a series of integral methods for the calculation of the conducting-gas flow in a high-current electric arc. The development of integral methods is of interest in that they are usually associated with adequate accuracy in determining integral values and values averaged over the cross section by a relatively simple calculation, and also allow the solution of the problem to be obtained in a number of situations when the realization of a difference method is complicated. The results of different calculation methods are compared. The effect of conditions in the initial cross section of the calculation region of the arc on its characteristics is investigated and a numerical analysis of the heating and acceleration of conducting gas is carried out.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 103–110, September–October, 1978.